After 38 days, desperate efforts to “plug the damn hole” seem to be making headway, as BP last night resumed pumping mud into the gushing oil well beneath the Gulf of Mexico.

The complex Top Kill operation is considered the best chance to halt the worst oil spill in US history.

Late Wednesday afternoon, BP began the operation — which involves pumping heavy drilling mud into the pipes and using robots to cork a blown-out oil well a mile below the surface.

But the effort stopped later that night to assess how things were going and bring in more materials. Pumping resumed last night.

The company also may shoot material into a crippled piece of equipment atop the well to keep the mud from escaping.

But BP, operator of the oil rig whose deadly explosion sparked the catastrophe, said it could be late today or the weekend before it knows whether the gusher has been capped. If Top Kill fails, the massive flow of gunk into coastal waters could last until August.

BP, which faces tens of billions of dollars in damages, has been under mounting pressure to “plug the damn hole” — as President Obama put it to an aide — shortly after the rig exploded on April 20, killing 11 workers.

The key to Top Kill is forcing mud into the gushing well and then filling it with cement, to seal it permanently.

BP managing director Robert Dudley compared the battle between the downward moving mud and the upward gushing oil to a “titanic arm wrestling match.”

Coast Guard Adm. Thad Allen — widely hailed for his performance in the aftermath of Hurricane Katrina — said yesterday that the mud was stopping some oil and gas.

“It’s a work in progress,” Allen told reporters in Venice, La. “We need to let it play itself out.”

There was also bad news yesterday.

A team of scientists concluded that the oil has been flowing at more than twice — and possibly up to five times — the rate previously thought.

Even using the most conservative estimate, that means the leak has grown to nearly 19 million gallons, nearly double the 11 million gallons of the record 1989 Exxon Valdez disaster in Alaska.

“Now we know the true scale of the monster we are fighting in the Gulf,” said Jeremy Symons, vice president of the National Wildlife Federation.